Forschung
Habilitation
- Uniform Error Estimation for Convection-Diffusion Problems, TU-Dresden, 2014,
mentor: Prof. H.-G. Roos,
arXiv:1403.0407
Doktorarbeit
- Singularly perturbed problems with characteristic layers: Supercloseness and postprocessing, TU-Dresden, 2008,
- supervisor: Jun.-Prof. T. Linß
Diplomarbeit
- Entwicklung eines impliziten LES Modells für kompressible Turbulenz, TU-Dresden, 2004,
supervisors: Prof. H.-G. Roos, Prof. N.A. Adams,
in German
Veröffentlichungen in Zeitschriften
- S. Franz, E. Yalçın:
A numerical method with the posteriori error estimates for fractional integro-differential equations.
accepted for publication in Int. J. Appl. Comput. Math. - A. Buchinger, S. Franz, N. Skrepek, and M. Waurick:
Homogenisation for Maxwell and Friends.
accepted for publication in SIAM Multiscale Modeling and Simulation, 2025. arXiv:2409.07084 - M. Bause, S. Franz:
Structure preserving discontinuous Galerkin approximation of a hyperbolic-parabolic system.
ETNA, 63, (2025), 1-32, arXiv:2311.01264 - S. Franz, N. Kopteva:
On the solution existence for collocation discretizations of time-fractional subdiffusion equations.
Journal of Scientific Computing, 100, (2024), article no. 68 - S. Franz, K. Liotati:
Numerical analysis of a singularly perturbed 4th order problem with a shift term.
APNUM, 201, (2024), 514--530, arxiv:2309.12105 - M. Brdar, S. Franz, H.-G. Roos:
A convection-diffusion problem with a large shift on Duran meshes.
Calcolo, 61(6), (2024), article no. 6, arXiv:2304.10937 - S. Franz:
Post-processing and improved error estimates of numerical methods for evolutionary systems.
IMA J. Numer. Anal., 44(5), (2024), 2936-2958,, arXiv:2304.12816 - S. Franz, N. Kopteva:
Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations.
J. Comp. Appl. Math., 427, (2023), article no. 115122, arXiv:2211.06272 - A. Demlow, S. Franz, N. Kopteva:
Maximum norm a posteriori error estimates for convection-diffusion problems.
IMA J. Numer. Anal., 43(5), (2023), 2562-2584, arXiv:2207.08251 - M. Brdar, S. Franz, L. Ludwig, H.-G. Roos:
Numerical analysis of a singularly perturbed convection diffusion problem with shift in space.
Appl. Numer. Math., 186, (2023), 129-142 - M. Brdar, S. Franz, L. Ludwig, H.-G. Roos:
A balanced norm error estimation for the time-dependent
reaction-diffusion problem with shift in space
Appl. Math. Comp., 437, (2023), article no. 127507 - Chr. Xenophontos, S. Franz, E. Sykopetritou :
Mixed hp FEM for singularly perturbed fourth order boundary value problems with two small parameters
Numer. Methods Partial Differential Equations, 38(5), (2022), 1543-1555 - S. Franz:
Singularly perturbed reaction-diffusion problems as first order systems,
J. Sci. Comput., 89, (2021), article no. 38 - S. Franz:
Anisotropic $H_{div}$-norm error estimates for rectangular $H_{div}$-elements,
Appl. Math. Letters, 121, (2021), 107453 - S. Franz:
Continuous time integration for changing type systems. - S. Franz, H.-G. Roos:
Error estimates in balanced norms of finite element methods for higher order reaction-diffusion problems. - M. Brdar, S. Franz, H.-G. Roos:
Numerical treatment for singularly perturbed fourth-order two-parameter problems,
ETNA, 51, (2019), 50-62 - P. Constantinou, S. Franz, L. Ludwig, Chr. Xenophontos:
A mixed hp FEM for the approximation of 4th order singularly perturbed problems on smooth domains
Numer. Methods Partial Differential Equations, 35(1), (2019), 114-127 - S. Franz, S. Trostorff, M. Waurick:
Numerical methods for changing type systems
IMA J. Numer. Anal., 39(2), (2019), 1009-1038 - L. Teofanov, M. Brdar, S. Franz, H. Zarin:
SDFEM for a singularly perturbed problem with two parameters.
Calcolo, 55, (2018), article no. 50 - P. Constantinou, S. Franz, L. Ludwig, Chr. Xenophontos:
Finite element approximation of reaction-diffusion problems using an exponentially graded mesh
Computer Math. Appl., 76, (2018), 2523-2534 - S. Franz, M. Waurick:
Resolvent estimates and numerical implementation for the homogenisation of one-dimensional periodic mixed type problems
ZAMM, 98(7), (2018), 1284-1294 - S. Franz, G. Matthies:
A Unified Framework for Time-Dependent Singularly Perturbed Problems with Discontinuous Galerkin in Time
Math. Comp., 87(313), (2018), 2113-2132 - S. Franz, K. Höhne, G. Matthies:
Grad-Div stabilisation applied to inf-sup stable discretisations with S-type meshes for the Oseen problem
IMA J. Numer. Anal., 38(1), (2018), 299-329 - S. Franz, K. Höhne, M. Waurick:
A solution decomposition for a singularly perturbed 4th order problem
Analysis, 37(2), (2017), 77-100 - S. Franz, Chr. Xenophontos:
A Short Note on the connection between layer-adapted exponentially graded and S-type meshes
CMAM, 18(2), (2017), 199-202 - S. Franz:
Convergence of LPS-FEM for convection-diffusion problems on layer-adapted meshes
BIT Numerical Mathematics, 57(3), (2017), 771-786 - S. Franz:
Analysis for a family of continuous-discontinuous Galerkin FEM for convection-diffusion problems.
Appl. Numer. Math., 110, (2016), 93-109 - S. Franz, L. Ludwig, Chr. Xenophontos:
Finite element approximation of convection-diffusion problems using an exponentially graded mesh
Computer Math. Appl., 72(6), (2016), 1532-1540 - S. Franz, H.-G. Roos:
Robust error estimation in energy and balanced norms for singularly perturbed forth order problems
Computer Math. Appl., 72(1), (2016), 233-247 - S. Franz, H.-G. Roos:
Error estimation in a balanced norm for a convection-diffusion problem with two different boundary layers.
Calcolo, 51(3), (2014), 423-440 , Addendum - S. Franz, H.-G. Roos:
Superconvergence of a Galerkin FEM for Higher-Order Elements in Convection-Diffusion Problems.
Numer. Math. Theor. Meth. Appl., 7(3), (2014), 356-373 , arXiv:1307.7543 - S. Franz, H.-G. Roos, A. Wachtel:
A C0 interior penalty method for a singularly-perturbed fourth-order elliptic problem on a layer-adapted mesh.
Numer. Methods Partial Differential Equations, 30(3) (2014), 838-861 - S. Franz:
Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems.
Appl. Numer. Math., 76 (2014), 132-144 , arXiv:1304.7443
Erratum on postprocessing operator - S. Franz:
Convergence Phenomena of Q_p-Elements for Convection-Diffusion Problems.
Numer. Methods Partial Differential Equations, 29(1), (2013), 280-296 - S. Franz, R. Gärtner, H.-G. Roos and A. Voigt:
A Note on the Convergence Analysis of a Diffuse-Domain Approach.
CMAM, 12(2), (2012), 153-167 - S. Franz, R. B. Kellogg, M. Stynes:
Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities.
Math. Comp., 81(278), (2012), 661-685 - S. Franz, N. Kopteva:
Green's function estimates for a singularly perturbed convection-diffusion problem.
Journal of Differential Equations, 252, (2012), 1521-1545 - S. Franz, N. Kopteva:
Green's function estimates for a singularly perturbed convection-diffusion problem in three dimensions.
Int. J. Numer. Anal. Model. Ser. B, 2(2-3), (2011), 124-141 - S. Franz, G. Matthies:
Convergence on Layer-adapted Meshes and Anisotropic Interpolation Error Estimates of Non-Standard Higher Order Finite Elements.
Appl. Numer. Math., 61 (2011), 723-737 , corrected preprint - S. Franz:
SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers.
BIT Numerical Mathematics, 51(3), (2011), 631-651 - S. Franz, H.-G. Roos:
The Capriciousness of Numerical Methods for Singular Perturbations.
SIAM Review, 53(1), (2011), 157-173 - S. Franz, G. Matthies:
Local projection stabilisation on S-type meshes for convection-diffusion problems with characteristic layers.
Computing, 87(3-4), (2010), 135-167 - S. Franz, T. Linß, H.-G. Roos, S. Schiller:
Uniform superconvergence of a finite element method with edge stabilization for convection-diffusion problems.
J. Comp. Math., 28(1), (2010), 32-44 - S. Franz, L. Tobiska, H. Zarin:
A New Approach to Recovery of Discontinuous Galerkin.
J. Comp. Math., 27(6), (2009), 697-712 - S. Franz, F. Liu, H.-G. Roos, M. Stynes, A. Zhou:
The combination technique for a two-dimensional convection-diffusion problem with exponential layers.
Appl. Math., 54(3), (2009), 203-223 - S. Franz, T. Linß, H.-G. Roos:
Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers,
Appl. Numer. Math., 58(12), (2008), 1818-1829 - S. Franz:
Continuous interior penalty method on a Shishkin mesh for convection-diffusion problems with characteristic boundary layers,
Comput. Meth. Appl. Mech. Engng., 197(45-48), (2008), 3679-3686 - S. Franz, T. Linß:
Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problem with characteristic layers,
Numer. Methods Partial Differential Equations, 24(1), (2008), 144-164 - N.A. Adams, S. Hickel, S. Franz:
Implicit subgrid-scale modeling by adaptive deconvolution,
J. Comput. Phys. 200(2), (2004), 412-431
Veröffentlichungen in Proceedings
- S. Franz, N. Kopteva:
Time stepping adaptation for subdiffusion problems with non-smooth right-hand sides.
accepted for publication in ENUMATH 2023 Proceedings, 2025, arxiv:2401.01586 -
U. Feldmann, S. Franz:
Above all: Motivation – some parts of our Maths for Engineers teaching concept.
In PAMM, 23(3), (2023) - J. Orwat, S. Franz:
Average graph of land surface curvatures induced mining operation determined by use of fourth order polynomial.
In AIP Conference Proceedings, 2293(1), (2020), article no. 350009. -
S. Franz, M. Waurick:
Homogenisation of parabolic/hyperbolic media.
In Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2018, Lecture Notes in Computational Science and Engineering 135 (2020), 215–229. Springer, Berlin Heidelberg - S. Franz, H.-G. Roos:
On robust error estimation for singularly perturbed fourth-order problems
In Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016
Lecture Notes in Computational Science and Engineering 120 (2017), 77-85, Springer, Berlin Heidelberg - S. Franz, H.-G. Roos:
Robust error estimation in energy and balanced norms for singularly perturbed forth order problems
Model. Anal. Inform. Sist., 23(3), (2016), 364-369 - S. Franz, J. Rubisch:
A discontinuous Galerkin method for fractional differential equations
PAMM, 16, (2016), 761-762 - S. Franz, N. Kopteva:
On the Sharpness of Green's function estimates for a convection-diffusion problem.
In Proceedings of the Fifth Conference on Finite Difference Methods: Theory and Applications (FDM: T&A 2010), (2011), 44-57, Rousse University Press, arXiv:1102.4520v2 - S. Franz, R. B. Kellogg, M. Stynes:
On the choice of mesh for a singularly perturbed problem with a corner singularity.
In BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods
Lecture Notes in Computational Science and Engineering 81, (2010), 119-126, Springer, Berlin Heidelberg - S. Franz, G. Matthies:
Local projection stabilisation on layer-adapted meshes for convection-diffusion problems with characteristic layers (Part I and II)
In BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods
Lecture Notes in Computational Science and Engineering 81, (2010), 127-138, Springer, Berlin Heidelberg - S. Franz, T. Linß:
Superconvergence analysis of Galerkin FEM and SDFEM for Elliptic Problems with characteristic layers,
Proceedings in Applied Mathematics and Mechanics 6, (2006) 1, 737-738. - N.A. Adams, S. Franz, S. Hickel, P. Koumoutsakos:
Optimization of an implicit subgrid-scale model for LES,
In Proceedings of the 21st International Congress of Theoretical and Applied Mechanics, Warsaw, Poland, 2004
Preprints und eingereichte Arbeiten
- S. Franz, S. Trostorff:
Numerical Treatment of Non-local Integral Operators in the Framework of Evolutionary Equations.
arXiv:2601.11132 - S. Franz, N. Kopteva:
Full analysis of the Green's function for a singularly perturbed convection-diffusion problem in three dimensions.
arXiv:1103.2948v2 - S. Franz:
A recovery operator for the combination method.
Preprint MATH-NM-09-2008, 10/2008, preprint